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					Project 2-Guidelines
 Recall- Class Project-Goals

 Determine what would be expected to happen if each
company bid the same amount as its signal.
 Determine the Company 1 bid under several
uniform bidding strategies, and explore the expected
values of these plans.
 Find a stable uniform bidding strategy that could be
followed by all companies, without any chance for
improvement.
  Recall-Project Assumptions
Assumption 1.
The same 18 companies
           will each bid on future similar leases
           only bidders for the tracts.
Assumption 2.
The geologists employed by companies
  equally expert
  on average, they can estimate the correct
  values of leases.
  each signal for the value of an undeveloped tract is
  an observation of a continuous random variable, Sv,

Mean of Sv  v ( actual value of lease )
   Recall-Project Assumptions
      Assumption 3. Except for their means, the
  distributions of the Sv’s are all identical
(The shape /The Spread)


  Assumption 4.
  All of the companies have the same profit margins
   Strategies for bidding on an Oil
                Lease
• Strategy 1
-Bid your signal. What will happen? Give reasoning for your
   analysis
• Strategy 2(First Plan)
-Subtract Winner’s curse from your signal to obtain bid.
   Assume all other companies do the same process. What
   will happen? Give reasoning for your analysis
• Strategy 3(Second Plan)
-Subtract Winner’s curse and Winner’s blessing from your
   signal to obtain bid. Assume all other companies do the
   same process. What will happen? Give reasoning for
   your analysis
   Strategies for bidding on an Oil
                Lease
• Strategy 4
-Find a optimal adjustment for company 1. Assume all
   other companies Subtract Winner’s curse and Winner’s
   blessing from their signals to obtain their bids.
• Strategy 5
-Find a optimal adjustment for company 1. Assume all
   other companies Subtract Winner’s curse from their
   signals to obtain their bids.

• Strategy 6
-Determine a stable Nash equilibrium bid. This stable
  strategy is such that any company will not have any
  incentive to deviate from
                        Strategy 1
-Bid your signal. What will happen? Give reasoning for your
                           analysis

•  The company (highest signal) will submit the highest bid-> win the
   lease.
• Under this plan, our company would submit a bid of its signal, s1 =
   $121,600,000
• “Winner’s Extra Profit” is almost always negative. That is, the winning
   company will not get its needed fair return on the lease.
• The mean of the 18 signals for a new lease = the actual value of the
   lease.Because for example, historical auction 1 the mean lease
   signal=85.3 M & proven value =91M
  (Assumption 2)


•    Hence, the highest signal will almost always be well above the value
    of the lease and the winning company will have paid too much for the
    drilling rights. This is called the winner’s curse.

•   WC=avg of the maximum error column
                  Strategy 2(First Plan)
  -Subtract Winner’s curse from your signal to obtain bid.
 Assume all other companies do the same process. What
      will happen? Give reasoning for your analysis
• defeat the winner’s curse.
• estimate the expected size of the curse
• Let C be the continuous random variable which gives the largest
  number in a sample of 18 observations of R(error).
• E(C)=winner’s curse=avg. of the maximum error column
• If a company bids its signal and wins the auction, it can expect,
  on average, to fall $22,600,000 below its needed fair return on
  the lease.
• If each company bids 22.6 million dollars less than its signal,
  then any one of the companies is equally likely to win the
  auction, and there will be no winner’s curse.
• Under this plan, our company would submit a bid of 121.6  22.6
  million dollars, that is $99,000,000
                       Strategy 3(Second Plan)
  -Subtract Winner’s curse and Winner’s blessing from your signal to
obtain bid. Assume all other companies do the same process. What will
               happen? Give reasoning for your analysis

• In an auction where the highest bid wins, any amount that the
  winner pays above the second highest bid is wasted.
• Let B be the continuous random variable which gives the difference
  between the largest and second largest errors in a random set of 18
  observations of R.

• E(B)=winners blessing=average of the difference column=5.8M
• the winning company will, on average, pay an unnecessary premium
  of 5.8 million.
• This leads to another possible bidding strategy, that we will call the
  Second Plan.
• Each company could bid $22,600,000 + $5,800,000 =
  $28,400,000 less than its signal.
• Under this plan, our company would submit a bid of 121.6  28.4
  million dollars, that is 93.2M
                                                             Bidding on
                                        Probability, Mathematics, Tests, Homework, Computers
                                                           an adjustment,
             The larger the expected value of Company 1’sOil Leasethe
    better is the long term effect of that bidding plan for us.
     on the project
              reduce by both the winner’s curse and blessing for Company 1 and
    all other companies.

     each company has approximately the same chance of winning, and, if a
    company wins, it can expect an extra profit that is close to the 5.8 million
    dollar winner’s blessing.                       Results Of Adjustment
                                                                    Average
                                                                     For All
                                                                     Other
                                                   Company 1 Companies
                         Probability of Winning        0.057          0.055
           Mean Extra Profit If Company Wins           5.843          5.832
               Expected Value Of Adjustment            0.335          0.323

Auction Focus.xls
 Simulating, Focus    Class Project     (material continues)             T    C     I    
                         Strategy 4
   -Find a optimal adjustment for company 1. Assume all
  other companies Subtract Winner’s curse and Winner’s
       blessing from their signals to obtain their bids       .
• steps
• 1. Enter the sum of the WC & WB as the signal adjustment for all
  other companies cell
• 2. Change company 1 signal adjustment cell to get a set of
  expected values for company 1.
• 3. record results of expected value for company 1
• How to find good adjustment points(10 points ) for company1?
• For example, If your WC=23,you could use 25,27,29,31,33
• and 21,19,17,15
• 4. enter each good adj. ->hit F9(to recalculate)->manually record
  the expected values & create a table for company 1
       Strategy 4-
                                              Bidding on
                         Probability, Mathematics, Tests, Homework, Computers
       Constructing      f(a) function Leasean Oil
on the project
                                      Company 1
for Company 1 must
                         Signal Adjustment:
find the maximum              Subtract
                                            Expected Value
expected value of                13             0.245
adjustment(assuming              15             0.434
that all other                   17             0.505
companies subtract               19             0.507
                                 21             0.511
both the curse and
                                 23             0.491
blessing)                        25             0.426
 this best adjustment,           27             0.373
acb                              29             0.327
                                 31             0.237
Strategy 4
                                                                     f(a) function
                         COMPANY 1: CURSE & BLESSING FOR ALL OTHERS
                   0.6
                   0.5
  Expected Value




                   0.4
                   0.3
                   0.2
                   0.1
                   0.0
                         0   2   4   6   8 10 12 14 16 18 20 22 24 26 28 30 32
                                              Signal Adjustment
copy from the sheet Strategy in Auction Focus.xls.
 Let f(a) be the expected value for Company 1 for subtracting a million
dollars from its signal, assuming that all other companies adjust their signals
by both the curse and blessing.
Fit a 4th degree polynomial trend line, which we will use as an approximate
                                                  Computation
formula for the unknown function f.            a     f (a )   f '(a )
Use solver to find the best adjustment 18.9166 0.524 0.000
                 Strategy 4
• This is the real world of business,competitors
  may also elect to subtract less than 28.4 million
  dollars from their signals.
•     there is a strong incentive for individual
  companies to deviate from the strategy and
  subtract less than the curse and blessing.
•     Company 1’s best adj. 18.9166 million is
  not itself a stable strategy.
                         Strategy 5
   -Find a optimal adjustment for company 1. Assume all
    other companies Subtract Winner’s curse from their
                  signals to obtain their bids    .
• steps
• 1. Enter the WC as the signal adjustment for all other companies
  cell
• 2. Change company 1 signal adjustment cell to get a set of
  expected values for company 1.
• 3. record results of expected value for company 1
• How to find good adjustment points(10 points ) for company1?
• For example,If your WC=23,you could use 25,27,29,31,33
• and 21,19,17,15
• 4. enter each good adj. ->hit F9(to recalculate)->manually record
  the expected values & create a table for company 1
      Strategy 5           Bidding on
                                        Probability, Mathematics,   Tests, Homework, Computers
                          an Oil Lease
      Constructing g(a) function
     on the project
                                                    Company 1
     for Company 1 must                Signal Adjustment:
                                                          Expected Value
     find the maximum                       Subtract
     expected value of                         17             -0.190
                                               19             -0.106
     adjustment(assuming
                                               21             -0.036
     that all other                            23             0.007
     companies subtract                        25             0.020
     both the curse and                        27             0.033
     blessing)                                 29             0.029
      this best adjustment,                    31             0.023
                                               33             0.018
     ac
                                               35             0.011
                                               37             0.007
Auction Focus.xls
  Simulating, Focus   Class Project                                     T      C     I     
                                      (material continues)
                               Strategy 5
                                                                        g(a) function
                     COMPANY 1: CURSE ONLY FOR ALL OTHERS
                  0.1
                  0.1
 Expected Value




                  0.0

                  -0.1 0   5    10    15    20    25     30   35   40

                  -0.1
                  -0.2
                  -0.2
                                     Signal Adjustment

USE the sheet Strategy in Auction Focus.xls.
 Let g(a) be the expected value for Company 1 for subtracting a million dollars from its signal,
assuming that all other companies adjust their signals by curse.
Fit a 4th degree polynomial trend line, which we will use as an approximate formula for the
unknown function g.
Use solver to find the best adjustment
 the use of Solver in Strategy shows that g(27.5990) = 0. Hence, ac = 27.5990 million
dollars.
            Company 1’s best response to an adjustment of 22.6 million dollars by all
other companies is to lower its signal by the considerably larger amount of $27,599,000.
                 Strategy 5
if we know what all other companies plan to
do. Moreover, this same information is available
to all of the bidders.
Need a stable strategy???
If all companies made such a stable
adjustment to their signals, then there would
be no incentive for anyone to alter the
strategy. A stable bidding strategy is also called
a Nash equilibrium
                       Strategy 6
How?
(a) Use Auction Equilibrium.xls(.
(b) FOLLOW THE INSTRUCTIONS IN THIS FILE!
(c) Enter appropriate values in cells B10 through E10.
(d) Enter a logical value in cell E39. Run the macro Optimize.
(the first logical value to use- (2wc+wb)/2
(e) Enter another logical value in cell E39 and press the key F9..
       record numbers in a table.
(f)    See table
(g) Find the stable adj for strategy 6
     MUST Download new file
• http://business.math.arizona.edu/MBD/mbd.html
• Link on class webpage
                                                   first logical value to
                                                   use for class

                        Strategy 6                 project-
                                                   (2wc+wb)/2=25.5
     Company 1           All Other     New logical value
     Optimal             Companies
     Adjustment, amax    Adjustment
     (use 4 decimals)    Subtracted
                         From Signal

1                                      (25+22.5414)/2=23.770
     22.5414             25.5
                                       7
2
                         23.7707
.

.

10

     Avg of amax=final
     stable adj for
     strategy 6
                                    Extra Profit
 • Extra profit is the amount by which the winning
   bid is below the fair value of a lease.
 • Let Xi be the random variable giving the extra
   profit gained by Company i. The sample of
   10,000 simulated auctions is used to
   approximate E(X1) and the average of E(X2),
   E(X3), ..., and E(X18)                  Results Of Adjustment
                                                                                  Average
Extra profit of winning company                                                   For All
                                                                                   Other
Company 1        Other Companies
                                                                       Company 1 Companies
                            -2.0
         -11.0                                  Probability of Winning   0.123     0.052
                              0.2   Mean Extra Profit If Company Wins    4.203     5.487
                             -3.7       Expected Value Of Adjustment     0.517     0.283

				
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posted:9/7/2011
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